AbstractThe binary interaction of tropical cyclones is investigated using a three‐dimensional primitive‐equation model. The extended anticyclonic circulations in the upper troposphere merge at very large vortex‐separation distances. For the cyclonic component in the lower troposphere, we find three fundamental modes of interaction separated by two critical separation distances: a mutual approach separation (MAS) and a mutual merger separation (MMS). We suggest that failure to identify these modes may have caused some confusion in interpreting previous baroclinic interaction studies.The MAS delineates vortices that approach each other from those that move on divergent orbits. The approach phase consists of steady radial movement and gradual acceleration, with deformation of the outer vorticity structure of each vortex but little change to their cores. In contrast to barotropic studies, the MAS is much larger than the radius at which the potential‐vorticity gradient of each vortex changes sign. Vortex tilting associated with the vertical shear of the azimuthal winds from the opposing vortex and secondary circulations associated with diabatic heating increases the mutual vortex attraction. The presence of an earth‐vorticity gradient reduces this attraction slightly, but also introduces considerable sensitivity to vortex orientation. When all physical processes are included, we find an MAS of around 1000 km with a scatter of several hundred km, which agrees well with observational studies.Approach occurs with little change in the vortex cores until they reach the MMS. Merger then occurs very rapidly, usually within several hours, and follows that described in parts II and III for barotropic vortices. The MMS is approximately three times the equivalent vortex‐patch radius for the cyclones; it is slightly reduced by diabatic heating, but it is largely independent of the earth‐vorticity gradient. The vortices experience a slight weakening during the approach and initial merger stages. However, with diabatic heating, rapid intensification follows merger; such intensification may have implications for rapid development of trop